To register on our site and for the best user experience, please enable Javascript in your browser using these instructions. Of particular interest is the analysis of the Paradox of Identity and a discussion of three solutions: Russell's views wend their way through the work, serving as a foil to Frege. Appendices give the proofs of the first 68 propositions of Begriffsschrift in modern notation.

This book will be of interest to students and professionals in philosophy and linguistics. It will set the agenda on some topics for some time to come and will bring new life to specific issues which have been widely discussed in the last 20 years. Function and argument 3. Sense and reference 4. Frege's Begriffsschrift theory of identity 5. Concept and object 6. Names and descriptions 7. Thought, truth value and assertion 9. Through the quotation marks. This title is available for institutional purchase via Cambridge Core Cambridge Core offers access to academic eBooks from our world-renowned publishing programme.

Please register or sign in to request access. If you are having problems accessing these resources please email lecturers cambridge. It is likely that Frege was offered a position as full Professor, but turned it down to avoid taking on additional administrative duties. His new position was unsalaried, but he was able to support himself and his family with a stipend from the Carl Zeiss Stiftung , a foundation that gave money to the University of Jena, and with which Ernst Abbe was intimately involved.

Because of the unfavorable reception of his earlier works, Frege was forced to arrange to have volume II of the G rundgesetze published at his own expense. It was not until that Frege was able to make such arrangements.

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However, while the volume was already in the publication process, Frege received a letter from Bertrand Russell, informing him that it was possible to prove a contradiction in the logical system of the first volume of the G rundgesetze , which included a naive calculus for classes. For more information, see the article on " Russell's Paradox ".

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Frege was, in his own words, "thunderstruck". He was forced to quickly prepare an appendix in response. For the next couple years, he continued to do important work. However, around , probably due to some combination of poor health, the early loss of his wife in , frustration with his failure to find an adequate solution to Russell's paradox, and disappointment over the continued poor reception of his work, Frege seems to have lost his intellectual steam.

He produced very little work between and his retirement in However, he continued to influence others during this period. Russell had included an appendix on Frege in his Principles of Mathematics. It is from this that Frege came be to be a bit wider known, including to an Austrian student studying engineering in Manchester, England, named Ludwig Wittgenstein. Wittgenstein studied the work of Frege and Russell closely, and in , he wrote to both of them concerning his own solution to Russell's paradox.

Frege invited him to Jena to discuss his views. Wittgenstein did so in late The two engaged in a philosophical debate, and while Wittgenstein reported that Frege "wiped the floor" with him, Frege was sufficiently impressed with Wittgenstein that he suggested that he go to Cambridge to study with Russell--a suggestion that had profound importance for the history of philosophy.

Moreover, Rudolf Carnap was one of Frege's students from to , and doubtlessly Frege had significant influence on Carnap's interest in logic and semantics and his subsequent intellectual development and successes. However, these were not wholly new works, but later drafts of works he had initiated in the s. In , a year before his death, Frege finally returned to the attempt to understand the foundations of arithmetic.

However, by this time, he had completely given up on his logicism, concluding that the paradoxes of class or set theory made it impossible.

## Frege, Gottlob | Internet Encyclopedia of Philosophy

He instead attempted to develop a new theory of the nature of arithmetic based on Kantian pure intuitions of space. However, he was not able to write much or publish anything about his new theory. Frege died on July 26, at the age of At the time of his death, Frege's own works were still not very widely known. He did not live to see the profound impact he would have on the emergence of analytic philosophy, nor to see his brand of logic--due to the championship of Russell--virtually wholly supersede earlier forms of logic. However, in bequeathing his unpublished work to his adopted son, Alfred, he wrote prophetically, "I believe there are things here which will one day be prized much more highly than they are now.

Take care that nothing gets lost. Unfortunately, however, they were destroyed in an Allied bombing raid on March 25, Although Scholz had made copies of some of the more important pieces, a good portion of Frege's unpublished works were lost. Although he was a fierce, sometimes even satirical, polemicist, Frege himself was a quiet, reserved man. He was right-wing in his political views, and like many conservatives of his generation in Germany, he is known to have been distrustful of foreigners and rather anti-semitic.

Himself Lutheran, Frege seems to have wanted to see all Jews expelled from Germany, or at least deprived of certain political rights. This distasteful feature of Frege's personality has gravely disappointed some of Frege's intellectual progeny. Trained as a mathematician, Frege's interests in logic grew out of his interests in the foundations of arithmetic. Early in his career, Frege became convinced that the truths of arithmetic are logical, analytic truths, agreeing with Leibniz , and disagreeing with Kant , who thought that arithmetical knowledge was grounded in "pure intuition", as well as more empiricist thinkers such as J.

Mill , who thought that arithmetic was grounded in observation. In other words, Frege subscribed to logicism. His logicism was modest in one sense, but very ambitious in others. Frege's logicism was limited to arithmetic; unlike other important historical logicists, such as Russell, Frege did not think that geometry was a branch of logic. However, Frege's logicism was very ambitious in another regard, as he believed that one could prove all of the truths of arithmetic deductively from a limited number of logical axioms. Indeed, Frege himself set out to demonstrate all of the basic laws of arithmetic within his own system of logic.

Frege concurred with Leibniz that natural language was unsuited to such a task. Thus, Frege sought to create a language that would combine the tasks of what Leibniz called a " calculus ratiocinator " and " lingua characterica ", that is, a logically perspicuous language in which logical relations and possible inferences would be clear and unambiguous. Frege's own term for such a language, "Begriffsschrift" was likely borrowed from a paper on Leibniz's ideas written by Adolf Trendelenburg.

Although there had been attempts to fashion at least the core of such a language made by Boole and others working in the Leibnizian tradition, Frege found their work unsuitable for a number of reasons.

Boole's logic used some of the same signs used in mathematics, except with different logical meanings. Frege found this unacceptable for a language which was to be used to demonstrate mathematical truths, because the signs would be ambiguous. Boole's logic, though innovative in some respects, was weak in others. It was divided into a "primary logic" and "secondary logic", bifurcating its propositional and categorical elements, and could not deal adequately with multiple generalities.

It analyzed propositions in terms of subject and predicate concepts, which Frege found to be imprecise and antiquated. Frege saw the formulae of mathematics as the paradigm of clear, unambiguous writing. In order to make his logical language suitable for purposes other than arithmetic, Frege expanded the notion of function to allow arguments and values other than numbers.

He defined a concept Begriff as a function that has a truth-value, either of the abstract objects the True or the False, as its value for any object as argument. See below for more on Frege's understanding of concepts, functions and objects. The concept being human is understood as a function that has the True as value for any argument that is human, and the False as value for anything else.

Suppose that " H " stands for this concept, and " a " is a constant for Aristotle, and " b " is a constant for the city of Boston. Then " H a " stands for the True, while " H b " stands for the False. In Frege's terminology, an object for which a concept has the True as value is said to " fall under " the concept. The values of such concepts could then be used as arguments to other functions. In his own logical systems, Frege introduced signs standing for the negation and conditional functions.

His own logical notation was two-dimensional. However, let us instead replace Frege's own notation with more contemporary notation. Conjunction and disjunction signs could then be defined from the negation and conditional signs. Frege also introduced an identity sign, standing for a function whose value is the True if the two arguments are the same object, and the False otherwise, and a sign, which he called "the horizontal," namely "—", that stands for a function that has the True as value for the True as argument, and has the False as value for any other argument.

Variables and quantifiers are used to express generalities. Frege understands quantifiers as "second-level concepts". The distinction between levels of functions involves what kind of arguments the functions take.

In Frege's view, unlike objects, all functions are "unsaturated" insofar as they require arguments to yield values. But different sorts of functions require different sorts of arguments. Functions that take first-level functions as argument are called second-level functions. Those familiar with modern predicate logic will recognize the parallels between it and Frege's logic. Frege is often credited with having founded predicate logic. However, Frege's logic is in some ways different from modern predicate logic. As we have seen, a sign such as " H " is a sign for a function in the strictest sense, as are the conditional and negation connectives.

Frege's conditional is not, like the modern connective, something that flanks statements to form a statement. Rather, it flanks terms for truth-values to form a term for a truth-value. Moreover, Frege's logical system was second-order. In addition to quantifiers ranging over objects, it also contained quantifiers ranging over first-level functions.

Frege's logic took the form of an axiomatic system. In fact, Frege was the first to take a fully axiomatic approach to logic, and the first even to suggest that inference rules ought to be explicitly formulated and distinguished from axioms. He began with a limited number of fixed axioms, introduced explicit inference rules, and aimed to derive all other logical truths including, for him, the truths of arithmetic from them.

Frege's first logical system, that of the Begriffsschrift , had nine axioms one of which was not independent , one explicit inference rule, and also employed a second and third inference rule implicitly. It represented the first axiomatization of logic, and was complete in its treatment of both propositional logic and first-order quantified logic.

## The Philosophy of Gottlob Frege

Unlike Frege's later system, the system of the Begriffsschrift was fully consistent. It has since been proven impossible to devise a system for higher-order logic with a finite number of axioms that is both complete and consistent. In order to make deduction easier, in the logical system of the Grundgesetze , Frege used fewer axioms and more inference rules: In the case of concepts, their value-ranges were identified with their extensions.

While Frege did sometimes also refer to the extensions of concepts as " classes ", he did not conceive of such classes as aggregates or collections. They were simply understood as objects corresponding to the complete argument-value mappings generated by concepts considered as functions. Frege then introduced two axioms dealing with these value-ranges.

Most infamous was his Basic Law V, which asserts that the truth-value of the value-range of function F being identical to the value-range of function G is the same as the truth-value of F and G having the same value for every argument. If one conceives of value-ranges as argument-value mappings, then this certainly seems to be a plausible hypothesis. However, from it, it is possible to prove a strong theorem of class membership: Given that value-ranges themselves are taken to be objects, if the concept in question is that of being a extension of a concept not included in itself , one can conclude that the extension of this concept is in itself just in case it is not.

Therefore, the logical system of the Grundgesetze was inconsistent due to Russell's Paradox. See the entry on Russell's Paradox for more details. However, the core of the system of the Grundgesetze , that is, the system minus the axioms governing value-ranges, is consistent and, like the system of the Begriffsschrift , is complete in its treatment of propositional logic and first-order predicate logic.

Given the extent to which it is taken granted today, it can be difficult to fully appreciate the truly innovative and radical approach Frege took to logic. Frege was the first to attempt to transcribe the old statements of categorical logic in a language employing variables, quantifiers and truth-functions. Frege was the first to understand a statement such as "all students are hardworking" as saying roughly the same as, "for all values of x , if x is a student, then x is hardworking". This made it possible to capture the logical connection between statements such as "either all students are hardworking or all students are intelligent" and "all students are either hardworking or intelligent" for example, that the first implies the second.

In earlier logical systems such as that of Boole, in which the propositional and quantificational elements were bifurcated, the connection was wholly lost. Moreover, Frege's logical system was the first to be able to capture statements of multiple generality, such as "every person loves some city" by using multiple quantifiers in the same logical formula.

This too was impossible in all earlier logical systems. Indeed, Frege's "firsts" in logic are almost too numerous to list. We have seen here that he invented modern quantification theory, presented the first complete axiomatization of propositional and first-order "predicate" logic the latter of which he invented outright , attempted the first formulation of higher-order logic, presented the first coherent and full analysis of variables and functions, first showed it possible to reduce all truth-functions to negation and the conditional, and made the first clear distinction between axioms and inference rules in a formal system.

As we shall see, he also made advances in the logic of mathematics. It is small wonder that he is often heralded as the founder of modern logic. On Frege's "philosophy of logic", logic is made true by a realm of logical entities. Logical functions, value-ranges, and the truth-values the True and the False, are thought to be objectively real entities, existing apart from the material and mental worlds.

As we shall see below , Frege was also committed to other logical entities such as senses and thoughts. Logical axioms are true because they express true thoughts about these entities. Thus, Frege denied the popular view that logic is without content and without metaphysical commitment. Frege was also a harsh critic of psychologism in logic: While Frege believed that logic might prescribe laws about how people should think, logic is not the science of how people do think.

Logical truths would remain true even if no one believed them nor used them in their reasoning. If humans were genetically designed to use regularly the so-called "inference rule" of affirming the consequent, etc. What is true or false, valid of invalid, does not depend on anyone's psychology or anyone's beliefs. To think otherwise is to confuse something's being true with something's being-taken-to-be-true.

Frege was an ardent proponent of logicism, the view that the truths of arithmetic are logical truths. Perhaps his most important contributions to the philosophy of mathematics were his arguments for this view. He also presented significant criticisms against rival views. We have seen that Frege was a harsh critic of psychologism in logic. He thought similarly about psychologism in mathematics. Numbers cannot be equated with anyone's mental images, nor truths of mathematics with psychological truths.

Mathematical truths are objective, not subjective. Frege was also a critic of Mill's view that arithmetical truths are empirical truths, based on observation. Frege pointed out that it is not just observable things that can be counted, and that mathematical truths seem to apply also to these things. On Mill's view, numbers must be taken to be conglomerations of objects.

Frege rejects this view for a number of reasons. Firstly, is one conglomeration of two things the same as a different conglomeration of two things, and if not, in what sense are they equal? Secondly, a conglomeration can be seen as made up of a different number of things, depending on how the parts are counted.

One deck of cards contains fifty two cards, but each card consists of a multitude of atoms. There is no one uniquely determined "number" of the whole conglomeration. He also reiterated the arguments of others: He also argued against the Kantian view that arithmetic truths are based on the pure intuition of the succession of time.

His main argument against this view, however, was simply his own work in which he showed that truths about the nature of succession and sequence can be proven purely from the axioms of logic. Frege was also an opponent of formalism, the view that arithmetic can be understood as the study of uninterpreted formal systems. While Frege's logical language represented a kind of formal system, he insisted that his formal system was important only because of what its signs represent and its propositions mean.

The signs themselves, independently of what they mean, are unimportant. To suggest that mathematics is the study simply of the formal system, is, in Frege's eyes, to confuse the sign and thing signified. Frege suggests also that this confusion would have the absurd result that numbers simply are the numerals, the signs on the page, and that we should be able to study their properties with a microscope. Frege suggests that rival views are often the result of attempting to understand the meaning of number terms in the wrong way, for example, in attempting to understand their meaning independently of the contexts in which they appear in sentences.

If we are simply asked to consider what "two" means independently of the context of a sentence, we are likely to simply imagine the numeral "2", or perhaps some conglomeration of two things. Thus, in the Grundlagen , Frege espouses his famous context principle , to "never ask for the meaning of a word in isolation, but only in the context of a proposition.

However, sentences 1 and 2 differ in truth-value. Mendelsohn diagnoses the problem as that sentence 1 , while containing the name "Cicero", is nevertheless not about Cicero the man, i. Hence, Mendelsohn reformulates the principle regarding substitution of coreferential terms as follows p. However, there are at least two problems with Mendelsohn's revised formulation. The first is that it seems to take the relation of aboutness as an additional semantic primitive, distinct from reference.

Another difficulty of which Mendelsohn is aware, but acknowledges only in a footnote is that the revised formulation is still false for certain sentences that contain multiple occurrences of the same term in different contexts. In virtue of its first conjunct, 3 is about Cicero. Nevertheless, we cannot substitute "Tully" for "Cicero" in 3 in both occurrences without getting something false. Frege's principle of substitution for reference would be better stated as follows: Certain of Mendelsohn's other formulations, such as an analogous principle regarding expressions with the same sense p.

It should also be noted that Mendelsohn's informal discussion of the philosophical issues involved is excellent, and my complaints about his formulations do not stem from any disagreement on the fundamental issues or the correct interpretation of Frege. The remaining chapters of the book deal with varied topics, from quotation, to the hierarchy of indirect senses, to the nature of truth and assertion. The clearest and best argued of the chapters are those dedicated to Frege's early account of identity in his Begriffsschrift , and the following chapter on Frege's distinction between concepts and objects.

He also criticizes Frege for not making it clear whether it is the notion of an object or the notion of a proper name that assumes priority: The bulk of the chapter is dedicated to a detailed, even painstaking, reconstruction of Frege's argument for the infamous claim that: The argument is presented with much more detail and rigor than is usually found in the secondary literature.

Mendelsohn faults for Frege for taking the surface grammar of ordinary language in a sentence such as 4 at face value. If translated into a formal language, 4 would become: Mendelsohn complains that Frege nowhere attempted to explain the connection between the semantics of descriptions and the semantics of predicate expressions. While Frege never addresses this issue in particular, he does claim that the sense of a complex expression is composed of the senses of the parts. Hence Frege would hold that the sense of the predicate expression should be regarded as part of the sense of the description.

### Contradictions in Frege’s system.

Mendelsohn's account of Russell's theory of descriptions is for the most part standard, though there are places where his wording is sloppy, or his portrayal misleading. For example, he claims that Russell held that "nonexistent objects do not have any properties", which is a very misleading way to word Russell's contention that every sentence containing a non-referring wide-scope description is false. The reason that "The King of France is bald" is false, for Russell, is not that the King of France doesn't have any properties, but that there is no King of France.

Mendelsohn also attributes a Meinongian ontology of non-existent objects to pre-"On Denoting" Russell, an interpretation that has been out of favor among Russell scholars for the past decade or more. Mendelsohn also assimilates Russell's "concepts" with his "propositional functions", again contrary to developments in the recent secondary literature.

Most egregious, however, is his claim p.